In a nutshell, what are the trigonometric functions?

Originally the trigonometric functions were defined in the
context of a right triangle. A more general definition can
be given in terms of the unit circle. Using that approach
also avoids getting stuck on the notion that those functions
are defined only for angles up to 90 degrees.

So consider an angle t with its vertex at the
origin, the initial side being the positive part of the
horizontal axis, and the terminal side intersecting a circle
of radius 1 centered at the origin, at a point
(x(t),y(t)). Now, that's a mouthful! It's
illustrated, however, and hopefully clearly, in this
picture:

Clearly, the coordinates x and y are
determined uniquely by the angle t.

There are three primary trigonometric functions, the
sine, cosine, and tangent. They are abbreviated by
sin, cos, tan and are defined by:

The reciprocals of these functions have also been given
names, they are called the cosecant, secant, and
cotangent, respectively, (abbreviated csc, sec, cot
). Thus

Only the first three matter, the others can always be
replaced by the reciprocals of sin, cos and
tan as needed. Indeed most calculators have keys
only for these three functions.

Warning: You calculator is likely to have keys
labeled

Those are keys for the inverse trig functions, not
their reciprocal! Indeed. most calculators have keys only
for the first three functions.

It's helpful to understand how the graphs of functions
relate to their formulas. As an example, consider:

This is a scaled and shifted sine curve.
Specifically,

a is a positive real number, the amplitude
, i.e., the maximum value of the function.

f is a real number, the frequency of
the function, i.e., the number of cycles per unit
interval. (The reciprocal of the frequency is the
period or periodicity .)

p is the time shift, a positive value
of p shifts the graph p units to the
left.

Click anywhere in the following list to see the graphs of
such and similar functions. For each example you'll see a
page showing the graph and the
Maple
command that generated it. You can cut and paste that
command into your own Maple session if you like.